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The Method of Direct Integration Examples 2

Recall from The Method of Direct Integration page that if we have a differential equation in the form $\frac{dy}{dt} = ay + b$ where $a$ and $b$ are constants, then the general solution can be obtained by rewriting this differential equation as to get the lefthand side to be the derivative of a natural logarithm while in general, the solution is given as:

(1)

\begin{align} y = De^{at} - \frac{b}{a} \end{align}

Let's look at some examples of solving differential equations by direct integration.

Note that if $D = 0$ we get that $y = 9$ which is a solution to our differential equation. Also, $y = 0$ (which we initially omitted) is also a solution to our differential equation. Thus all solutions are given by $y = De^{4t} + 9$ for $D$ as a constant, and $y = 0$.